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Journal articles on the topic 'Mathematical modelling/biology'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Vasieva, Olga, Manan'Iarivo Rasolonjanahary, and Bakhtier Vasiev. "Mathematical modelling in developmental biology." REPRODUCTION 145, no. 6 (2013): R175—R184. http://dx.doi.org/10.1530/rep-12-0081.

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In recent decades, molecular and cellular biology has benefited from numerous fascinating developments in experimental technique, generating an overwhelming amount of data on various biological objects and processes. This, in turn, has led biologists to look for appropriate tools to facilitate systematic analysis of data. Thus, the need for mathematical techniques, which can be used to aid the classification and understanding of this ever-growing body of experimental data, is more profound now than ever before. Mathematical modelling is becoming increasingly integrated into biological studies in general and into developmental biology particularly. This review outlines some achievements of mathematics as applied to developmental biology and demonstrates the mathematical formulation of basic principles driving morphogenesis. We begin by describing a mathematical formalism used to analyse the formation and scaling of morphogen gradients. Then we address a problem of interplay between the dynamics of morphogen gradients and movement of cells, referring to mathematical models of gastrulation in the chick embryo. In the last section, we give an overview of various mathematical models used in the study of the developmental cycle of Dictyostelium discoideum, which is probably the best example of successful mathematical modelling in developmental biology.

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2

Tomlin, Claire J., and Jeffrey D. Axelrod. "Biology by numbers: mathematical modelling in developmental biology." Nature Reviews Genetics 8, no. 5 (2007): 331–40. http://dx.doi.org/10.1038/nrg2098.

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3

Jäger, Willi. "Mathematical Modelling in Chemistry and Biology." Interdisciplinary Science Reviews 11, no. 2 (1986): 181–88. http://dx.doi.org/10.1179/isr.1986.11.2.181.

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4

Chaplain, M. A. J. "Multiscale mathematical modelling in biology and medicine." IMA Journal of Applied Mathematics 76, no. 3 (2011): 371–88. http://dx.doi.org/10.1093/imamat/hxr025.

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5

Lei, Jinzhi. "Viewpoints on modelling: Comments on "Achilles and the tortoise: Some caveats to mathematical modelling in biology"." Mathematics in Applied Sciences and Engineering 1, no. 1 (2020): 85–90. http://dx.doi.org/10.5206/mase/10267.

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Mathematical modelling has been proven to be useful in understanding some problems from biological science, provided that it is used properly. However, it has also attracted some criticisms as partially presented in a recent opinion article \cite{Gilbert2018} from biological community. This note intends to clarify some confusion and misunderstanding in regard to mathematically modelling by commenting on those critiques raised in \cite{Gilbert2018}, with a hope of initiating some further discussion so that both applied mathematicians and biologist can better use mathematical modelling and better understand the results from modelling.

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Middleton, A., M. Owen, M. Bennett, and J. King. "Mathematical modelling of gibberellinsignalling." Comparative Biochemistry and Physiology Part A: Molecular & Integrative Physiology 150, no. 3 (2008): S46. http://dx.doi.org/10.1016/j.cbpa.2008.04.023.

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7

Butler, George, Jonathan Rudge, and Philip R. Dash. "Mathematical modelling of cell migration." Essays in Biochemistry 63, no. 5 (2019): 631–37. http://dx.doi.org/10.1042/ebc20190020.

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Abstract The complexity of biological systems creates challenges for fully understanding their behaviour. This is particularly true for cell migration which requires the co-ordinated activity of hundreds of individual components within cells. Mathematical modelling can help understand these complex systems by breaking the system into discrete steps which can then be interrogated in silico. In this review, we highlight scenarios in cell migration where mathematical modelling can be applied and discuss what types of modelling are most suited. Almost any aspect of cell migration is amenable to mathematical modelling from the modelling of intracellular processes such as chemokine receptor signalling and actin filament branching to larger scale processes such as the movement of individual cells or populations of cells through their environment. Two common ways of approaching this modelling are the use of models based on differential equations or agent-based modelling. The application of both these approaches to cell migration are discussed with specific examples along with common software tools to facilitate the process for non-mathematicians. We also highlight the challenges of modelling cell migration and the need for rigorous experimental work to effectively parameterise a model.

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Divya, B., and K. Kavitha. "A REVIEW ON MATHEMATICAL MODELLING IN BIOLOGY AND MEDICINE." Advances in Mathematics: Scientific Journal 9, no. 8 (2020): 5869–79. http://dx.doi.org/10.37418/amsj.9.8.54.

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9

MacArthur, B. D., C. P. Please, M. Taylor, and R. O. C. Oreffo. "Mathematical modelling of skeletal repair." Biochemical and Biophysical Research Communications 313, no. 4 (2004): 825–33. http://dx.doi.org/10.1016/j.bbrc.2003.11.171.

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10

Shone, John. "Working at the biology/mathematics interface: mathematical modelling and sixth form biology." International Journal of Mathematical Education in Science and Technology 19, no. 4 (1988): 501–9. http://dx.doi.org/10.1080/0020739880190402.

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11

Friedman, Avner. "Free boundary problems in biology." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373, no. 2050 (2015): 20140368. http://dx.doi.org/10.1098/rsta.2014.0368.

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In this paper, I review several free boundary problems that arise in the mathematical modelling of biological processes. The biological topics are quite diverse: cancer, wound healing, biofilms, granulomas and atherosclerosis. For each of these topics, I describe the biological background and the mathematical model, and then proceed to state mathematical results, including existence and uniqueness theorems, stability and asymptotic limits, and the behaviour of the free boundary. I also suggest, for each of the topics, open mathematical problems.

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12

Lampl, Michelle. "Perspectives on modelling human growth: Mathematical models and growth biology." Annals of Human Biology 39, no. 5 (2012): 342–51. http://dx.doi.org/10.3109/03014460.2012.704072.

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13

Eftimie, Raluca. "The quest for a new modelling framework in mathematical biology." Physics of Life Reviews 12 (March 2015): 72–73. http://dx.doi.org/10.1016/j.plrev.2015.01.012.

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14

Alfonso, J. C. L., K. Talkenberger, M. Seifert, et al. "The biology and mathematical modelling of glioma invasion: a review." Journal of The Royal Society Interface 14, no. 136 (2017): 20170490. http://dx.doi.org/10.1098/rsif.2017.0490.

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Adult gliomas are aggressive brain tumours associated with low patient survival rates and limited life expectancy. The most important hallmark of this type of tumour is its invasive behaviour, characterized by a markedly phenotypic plasticity, infiltrative tumour morphologies and the ability of malignant progression from low- to high-grade tumour types. Indeed, the widespread infiltration of healthy brain tissue by glioma cells is largely responsible for poor prognosis and the difficulty of finding curative therapies. Meanwhile, mathematical models have been established to analyse potential mechanisms of glioma invasion. In this review, we start with a brief introduction to current biological knowledge about glioma invasion, and then critically review and highlight future challenges for mathematical models of glioma invasion.

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15

Maini, P. K. "Essential Mathematical Biology." Mathematical Medicine and Biology 20, no. 2 (2003): 225–26. http://dx.doi.org/10.1093/imammb/20.2.225.

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16

Geris, L., J. Vander Sloten, and H. Van Oosterwyck. "In silico biology of bone modelling and remodelling: regeneration." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367, no. 1895 (2009): 2031–53. http://dx.doi.org/10.1098/rsta.2008.0293.

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Bone regeneration is the process whereby bone is able to (scarlessly) repair itself from trauma, such as fractures or implant placement. Despite extensive experimental research, many of the mechanisms involved still remain to be elucidated. Over the last decade, many mathematical models have been established to investigate the regeneration process in silico . The first models considered only the influence of the mechanical environment as a regulator of the healing process. These models were followed by the development of bioregulatory models where mechanics was neglected and regeneration was regulated only by biological stimuli such as growth factors. The most recent mathematical models couple the influences of both biological and mechanical stimuli. Examples are given to illustrate the added value of mathematical regeneration research, specifically in the in silico design of treatment strategies for non-unions. Drawbacks of the current continuum-type models, together with possible solutions in extending the models towards other time and length scales are discussed. Finally, the demands for dedicated and more quantitative experimental research are presented.

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Lewis, Rohan M., Jane K. Cleal, and Bram G. Sengers. "Placental perfusion and mathematical modelling." Placenta 93 (April 2020): 43–48. http://dx.doi.org/10.1016/j.placenta.2020.02.015.

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18

Huber, Heinrich J., Heiko Duessmann, Jakub Wenus, Seán M. Kilbride, and Jochen H. M. Prehn. "Mathematical modelling of the mitochondrial apoptosis pathway." Biochimica et Biophysica Acta (BBA) - Molecular Cell Research 1813, no. 4 (2011): 608–15. http://dx.doi.org/10.1016/j.bbamcr.2010.10.004.

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19

Alexander, R. McN. "Modelling approaches in biomechanics." Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 358, no. 1437 (2003): 1429–35. http://dx.doi.org/10.1098/rstb.2003.1336.

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Conceptual, physical and mathematical models have all proved useful in biomechanics. Conceptual models, which have been used only occasionally, clarify a point without having to be constructed physically or analysed mathematically. Some physical models are designed to demonstrate a proposed mechanism, for example the folding mechanisms of insect wings. Others have been used to check the conclusions of mathematical modelling. However, others facilitate observations that would be difficult to make on real organisms, for example on the flow of air around the wings of small insects. Mathematical models have been used more often than physical ones. Some of them are predictive, designed for example to calculate the effects of anatomical changes on jumping performance, or the pattern of flow in a 3D assembly of semicircular canals. Others seek an optimum, for example the best possible technique for a high jump. A few have been used in inverse optimization studies, which search for variables that are optimized by observed patterns of behaviour. Mathematical models range from the extreme simplicity of some models of walking and running, to the complexity of models that represent numerous body segments and muscles, or elaborate bone shapes. The simpler the model, the clearer it is which of its features is essential to the calculated effect.

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20

Ehrhardt, Matthias, Lucas Jódar Sánchez, and Rafael Jacinto Villanueva Micó. "Numerical methods and mathematical modelling in biology, medicine and social sciences." International Journal of Computer Mathematics 91, no. 2 (2014): 176–78. http://dx.doi.org/10.1080/00207160.2014.896653.

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21

Brunetti, Mia, Michael C. Mackey, and Morgan Craig. "Understanding Normal and Pathological Hematopoietic Stem Cell Biology Using Mathematical Modelling." Current Stem Cell Reports 7, no. 3 (2021): 109–20. http://dx.doi.org/10.1007/s40778-021-00191-9.

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22

Vera, Julio, Christopher Lischer, Momchil Nenov, Svetoslav Nikolov, Xin Lai, and Martin Eberhardt. "Mathematical Modelling in Biomedicine: A Primer for the Curious and the Skeptic." International Journal of Molecular Sciences 22, no. 2 (2021): 547. http://dx.doi.org/10.3390/ijms22020547.

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In most disciplines of natural sciences and engineering, mathematical and computational modelling are mainstay methods which are usefulness beyond doubt. These disciplines would not have reached today’s level of sophistication without an intensive use of mathematical and computational models together with quantitative data. This approach has not been followed in much of molecular biology and biomedicine, however, where qualitative descriptions are accepted as a satisfactory replacement for mathematical rigor and the use of computational models is seen by many as a fringe practice rather than as a powerful scientific method. This position disregards mathematical thinking as having contributed key discoveries in biology for more than a century, e.g., in the connection between genes, inheritance, and evolution or in the mechanisms of enzymatic catalysis. Here, we discuss the role of computational modelling in the arsenal of modern scientific methods in biomedicine. We list frequent misconceptions about mathematical modelling found among biomedical experimentalists and suggest some good practices that can help bridge the cognitive gap between modelers and experimental researchers in biomedicine. This manuscript was written with two readers in mind. Firstly, it is intended for mathematical modelers with a background in physics, mathematics, or engineering who want to jump into biomedicine. We provide them with ideas to motivate the use of mathematical modelling when discussing with experimental partners. Secondly, this is a text for biomedical researchers intrigued with utilizing mathematical modelling to investigate the pathophysiology of human diseases to improve their diagnostics and treatment.

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23

Vera, Julio, Christopher Lischer, Momchil Nenov, Svetoslav Nikolov, Xin Lai, and Martin Eberhardt. "Mathematical Modelling in Biomedicine: A Primer for the Curious and the Skeptic." International Journal of Molecular Sciences 22, no. 2 (2021): 547. http://dx.doi.org/10.3390/ijms22020547.

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In most disciplines of natural sciences and engineering, mathematical and computational modelling are mainstay methods which are usefulness beyond doubt. These disciplines would not have reached today’s level of sophistication without an intensive use of mathematical and computational models together with quantitative data. This approach has not been followed in much of molecular biology and biomedicine, however, where qualitative descriptions are accepted as a satisfactory replacement for mathematical rigor and the use of computational models is seen by many as a fringe practice rather than as a powerful scientific method. This position disregards mathematical thinking as having contributed key discoveries in biology for more than a century, e.g., in the connection between genes, inheritance, and evolution or in the mechanisms of enzymatic catalysis. Here, we discuss the role of computational modelling in the arsenal of modern scientific methods in biomedicine. We list frequent misconceptions about mathematical modelling found among biomedical experimentalists and suggest some good practices that can help bridge the cognitive gap between modelers and experimental researchers in biomedicine. This manuscript was written with two readers in mind. Firstly, it is intended for mathematical modelers with a background in physics, mathematics, or engineering who want to jump into biomedicine. We provide them with ideas to motivate the use of mathematical modelling when discussing with experimental partners. Secondly, this is a text for biomedical researchers intrigued with utilizing mathematical modelling to investigate the pathophysiology of human diseases to improve their diagnostics and treatment.

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24

McDonald, Andrew G., Keith F. Tipton, and Gavin P. Davey. "Mathematical modelling of metabolism: Summing up." Biochemist 31, no. 3 (2009): 24–27. http://dx.doi.org/10.1042/bio03103024.

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Biochemistry is as much a quantitative subject as qualitative. Initial observations of single- or multicellular organisms have given rise to our discipline, which is the discovery and characterization of the chemistry of all living things. We have moved from Lavoisier's seminal observation of respiration as a form of combustion, to a much more detailed knowledge of the associated biochemistry.

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25

Shelah, Saharon, and Lutz Strüngmann. "Infinite combinatorics in mathematical biology." Biosystems 204 (June 2021): 104392. http://dx.doi.org/10.1016/j.biosystems.2021.104392.

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26

Priori, Luca, and Paolo Ubezio. "Mathematical modelling and computer simulation of cell synchrony." Methods in Cell Science 18, no. 2 (1996): 83–91. http://dx.doi.org/10.1007/bf00122158.

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27

Ribba, Benjamin, Philippe Tracqui, Jean-Laurent Boix, Jean-Pierre Boissel, and S. Randall Thomas. "Q x DB: a generic database to support mathematical modelling in biology." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364, no. 1843 (2006): 1517–32. http://dx.doi.org/10.1098/rsta.2006.1784.

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Q x DB (quantitative x -modelling database) is a web-based generic database package designed especially to house quantitative and structural information. Its development was motivated by the need for centralized access to such results for development of mathematical models, but its usefulness extends to the general research community of both modellers and experimentalists. Written in PHP (Hyper Preprocessor) and MySQL , the database is easily adapted to new fields of research and ported to Apache-based web servers. Unlike most existing databases, experimental and observational results curated in Q x DB are supplemented by comments from the experts who contribute input to the database, giving their evaluations of experimental techniques, breadth of validity of results, experimental conditions, and the like, thus providing the visitor with a basis for gauging the quality (or appropriateness) of each item for his/her needs. Q x DB can be easily customized by adapting the contents of the database table containing the descriptors that characterize each data record according to an informal ontology of the research domain. We will illustrate this adaptability of Q x DB by presenting two examples, the first dealing with modelling in oncology and the second with mechanical properties of cells and tissues.

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28

Torres, Néstor V. "Introducing Systems Biology to Bioscience Students through Mathematical Modelling. A Practical Module." Bioscience Education 21, no. 1 (2013): 54–63. http://dx.doi.org/10.11120/beej.2013.00012.

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29

Auger, Pierre, and Jean-Christophe Poggiale. "Mathematical modelling is a necessary step in biology and in environmental sciences." Comptes Rendus Geoscience 338, no. 4 (2006): 223–24. http://dx.doi.org/10.1016/j.crte.2006.01.004.

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30

Prpic, Nikola-Michael, and Nico Posnien. "Size and shape—integration of morphometrics, mathematical modelling, developmental and evolutionary biology." Development Genes and Evolution 226, no. 3 (2016): 109–12. http://dx.doi.org/10.1007/s00427-016-0536-5.

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31

Wearing, H. "Mathematical Modelling of Juxtacrine Patterning." Bulletin of Mathematical Biology 62, no. 2 (2000): 293–320. http://dx.doi.org/10.1006/bulm.1999.0152.

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32

Smallbone, Kieran, Robert A. Gatenby, and Philip K. Maini. "Mathematical modelling of tumour acidity." Journal of Theoretical Biology 255, no. 1 (2008): 106–12. http://dx.doi.org/10.1016/j.jtbi.2008.08.002.

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33

Liebal, Ulf W., Thomas Millat, Imke G. De Jong, Oscar P. Kuipers, Uwe Völker, and Olaf Wolkenhauer. "How mathematical modelling elucidates signalling in Bacillus subtilis." Molecular Microbiology 77, no. 5 (2010): 1083–95. http://dx.doi.org/10.1111/j.1365-2958.2010.07283.x.

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34

Meades, G., N. K. Thalji, M. de Queiroz, X. Cai, and G. L. Waldrop. "Mathematical modelling of negative feedback regulation by carboxyltransferase." IET Systems Biology 5, no. 3 (2011): 220–28. http://dx.doi.org/10.1049/iet-syb.2010.0071.

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35

Rangamani, Padmini, and Ravi Iyengar. "Modelling cellular signalling systems." Essays in Biochemistry 45 (September 30, 2008): 83–94. http://dx.doi.org/10.1042/bse0450083.

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Cell signalling pathways and networks are complex and often non-linear. Signalling pathways can be represented as systems of biochemical reactions that can be modelled using differential equations. Computational modelling of cell signalling pathways is emerging as a tool that facilitates mechanistic understanding of complex biological systems. Mathematical models are also used to generate predictions that may be tested experimentally. In the present chapter, the various steps involved in building models of cell signalling pathways are discussed. Depending on the nature of the process being modelled and the scale of the model, different mathematical formulations, ranging from stochastic representations to ordinary and partial differential equations are discussed. This is followed by a brief summary of some recent modelling successes and the state of future models.

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36

Cho, K. H., and O. Wolkenhauer. "Analysis and modelling of signal transduction pathways in systems biology." Biochemical Society Transactions 31, no. 6 (2003): 1503–9. http://dx.doi.org/10.1042/bst0311503.

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There is general agreement that a systems approach is needed for a better understanding of causal and functional relationships that generate the dynamics of biological networks and pathways. These observations have been the basis for efforts to get the engineering and physical sciences involved in life sciences. The emergence of systems biology as a new area of research is evidence for these developments. Dynamic modelling and simulation of signal transduction pathways is an important theme in systems biology and is getting growing attention from researchers with an interest in the analysis of dynamic systems. This paper introduces systems biology in terms of the analysis and modelling of signal transduction pathways. Focusing on mathematical representations of cellular dynamics, a number of emerging challenges and perspectives are discussed.

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37

Ali Lashari, Abid, and Faiz Ahmad. "False mathematical reasoning in biology." Journal of Theoretical Biology 307 (August 2012): 211. http://dx.doi.org/10.1016/j.jtbi.2012.05.006.

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38

Westerhoff, Hans V. "Mathematical and theoretical biology for systems biology, and then ... vice versa." Journal of Mathematical Biology 54, no. 1 (2006): 147–50. http://dx.doi.org/10.1007/s00285-006-0043-9.

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39

Varner, J. D. "Systems biology and the mathematical modelling of antibody-directed enzyme prodrug therapy (ADEPT)." IEE Proceedings - Systems Biology 152, no. 4 (2005): 291. http://dx.doi.org/10.1049/ip-syb:20050047.

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40

Burrows, M. T. "Modelling biological populations in space and time: Cambridge studies in mathematical biology: 11." Journal of Experimental Marine Biology and Ecology 168, no. 1 (1993): 139–40. http://dx.doi.org/10.1016/0022-0981(93)90120-d.

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41

MUKHERJEE, SHIBAJI, and SUSHMITA MITRA. "HIDDEN MARKOV MODELS, GRAMMARS, AND BIOLOGY: A TUTORIAL." Journal of Bioinformatics and Computational Biology 03, no. 02 (2005): 491–526. http://dx.doi.org/10.1142/s0219720005001077.

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Biological sequences and structures have been modelled using various machine learning techniques and abstract mathematical concepts. This article surveys methods using Hidden Markov Model and functional grammars for this purpose. We provide a formal introduction to Hidden Markov Model and grammars, stressing on a comprehensive mathematical description of the methods and their natural continuity. The basic algorithms and their application to analyzing biological sequences and modelling structures of bio-molecules like proteins and nucleic acids are discussed. A comparison of the different approaches is discussed, and possible areas of work and problems are highlighted. Related databases and softwares, available on the internet, are also mentioned.

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42

Axenie, Cristian, Roman Bauer, and María Rodríguez Martínez. "The Multiple Dimensions of Networks in Cancer: A Perspective." Symmetry 13, no. 9 (2021): 1559. http://dx.doi.org/10.3390/sym13091559.

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This perspective article gathers the latest developments in mathematical and computational oncology tools that exploit network approaches for the mathematical modelling, analysis, and simulation of cancer development and therapy design. It instigates the community to explore new paths and synergies under the umbrella of the Special Issue “Networks in Cancer: From Symmetry Breaking to Targeted Therapy”. The focus of the perspective is to demonstrate how networks can model the physics, analyse the interactions, and predict the evolution of the multiple processes behind tumour-host encounters across multiple scales. From agent-based modelling and mechano-biology to machine learning and predictive modelling, the perspective motivates a methodology well suited to mathematical and computational oncology and suggests approaches that mark a viable path towards adoption in the clinic.

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Rittscher, Jens, Andrew Blake, Anthony Hoogs, and Gees Stein. "Mathematical modelling of animate and intentional motion." Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 358, no. 1431 (2003): 475–90. http://dx.doi.org/10.1098/rstb.2002.1259.

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Our aim is to enable a machine to observe and interpret the behaviour of others. Mathematical models are employed to describe certain biological motions. The main challenge is to design models that are both tractable and meaningful. In the first part we will describe how computer vision techniques, in particular visual tracking, can be applied to recognize a small vocabulary of human actions in a constrained scenario. Mainly the problems of viewpoint and scale invariance need to be overcome to formalize a general framework. Hence the second part of the article is devoted to the question whether a particular human action should be captured in a single complex model or whether it is more promising to make extensive use of semantic knowledge and a collection of low–level models that encode certain motion primitives. Scene context plays a crucial role if we intend to give a higher–level interpretation rather than a low–level physical description of the observed motion. A semantic knowledge base is used to establish the scene context. This approach consists of three main components: visual analysis, the mapping from vision to language and the search of the semantic database. A small number of robust visual detectors is used to generate a higher–level description of the scene. The approach together with a number of results is presented in the third part of this article.

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44

Owen, Markus R., and Jonathan A. Sherratt. "Mathematical modelling of juxtacrine cell signalling." Mathematical Biosciences 153, no. 2 (1998): 125–50. http://dx.doi.org/10.1016/s0025-5564(98)10034-2.

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45

Lemon, Greg, Daniel Howard, Matthew J. Tomlinson, et al. "Mathematical modelling of tissue-engineered angiogenesis." Mathematical Biosciences 221, no. 2 (2009): 101–20. http://dx.doi.org/10.1016/j.mbs.2009.07.003.

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46

Baigent, Stephen, Robert Unwin, and Chee Chit Yeng. "Mathematical Modelling of Profiled Haemodialysis: A Simplified Approach." Journal of Theoretical Medicine 3, no. 2 (2001): 143–60. http://dx.doi.org/10.1080/10273660108833070.

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For many renal patients with severe loss of kidney function dialysis treatment is the only means of preventing excessive fluid gain and the accumulation of toxic chemicals in the blood. Typically, haemodialysis patients will dialyse three times a week, with each session lasting 4-6 hours. During each session, 2-3 litres of fluid is removed along with catabolic end-products, and osmotically active solutes. In a significant number of patients, the rapid removal of water and osmotically active sodium chloride can lead to hypotension or overhydration and swelling of brain cells. Profiled haemodialysis, in which the rate of water removal and/or the dialysis machine sodium concentration are varied according to a predetermined profile, can help to prevent wide fluctuations in plasma osmolality, which cause these complications. The profiles are determined on a trial and error basis, and differ from patient to patient. Here we describe a mathematical model for a typical profiled haemodialysis session in which the variables of interest are sodium mass and body fluid volumes. The model is of minimal complexity and so could provide simple guidelines for choosing suitable profiles for individual patients. The model is tested for a series of dialysate sodium profiles to demonstrate the potential benefits of sodium profiling. Next, using the simplicity of the model, we show how to calculate the dialysate sodium profile to model a dialysis session that achieves specified targets of sodium mass removal and weight loss, while keeping the risk of intradia-lytic complications to a minimum. Finally, we investigate which of the model profiled dialysis sessions that meet a range of sodium and fluid removal targets also predict extracellular sodium concentrations and extracellular volumes that lie within “safe” limits. Our model suggests that improvements in volume control via sodium profiling need to be set against potential problems in maintaining blood concentrations and body fluid compartment volumes within “safe” limits.

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47

Marshall, James A. R., Andreagiovanni Reina, and Thomas Bose. "Multiscale Modelling Tool: Mathematical modelling of collective behaviour without the maths." PLOS ONE 14, no. 9 (2019): e0222906. http://dx.doi.org/10.1371/journal.pone.0222906.

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48

Alt, Wolfgang. "ESMTB – European Society for Mathematical and Theoretical Biology." Journal of Mathematical Biology 53, no. 3 (2006): 337–39. http://dx.doi.org/10.1007/s00285-006-0030-1.

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49

Deutsch, Andreas. "ESMTB-European Society for Mathematical and Theoretical Biology." Journal of Mathematical Biology 53, no. 5 (2006): 887–88. http://dx.doi.org/10.1007/s00285-006-0041-y.

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Diekmann, Odo, Klaus Dietz, Thomas Hillen, and Horst Thieme. "Karl-Peter Hadeler: His legacy in mathematical biology." Journal of Mathematical Biology 77, no. 6-7 (2018): 1623–27. http://dx.doi.org/10.1007/s00285-018-1259-1.

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